In this work, the behaviour of solutions for the Dirichlet problem of the non-local equation

$$ u_t=\varDelta(\kappa(u))+\frac{\lambda f(u)}{(\int_{\varOmega}f(u)\,\mathrm{d}x)^p},\quad \varOmega\subset\mathbb{R}^N,\quad N=1,2, $$

is studied, mainly for the case where $f(s)=\mathrm{e}^{\kappa(s)}$. More precisely, the interplay of exponent $p$ of the non-local term and spatial dimension $N$ is investigated with regard to the existence and non-existence of solutions of the associated steady-state problem as well as the global existence and finite-time blow-up of the time-dependent solutions $u(x,t)$. The asymptotic stability of the steady-state solutions is also studied.

AMS 2000 Mathematics subject classification: Primary 35K60. Secondary 35B40